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Complexity of Self-Assembled Shapes

Soloveichik, David and Winfree, Erik (2007) Complexity of Self-Assembled Shapes. SIAM Journal on Computing, 36 (6). pp. 1544-1569. ISSN 0097-5397. doi:10.1137/S0097539704446712.

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The connection between self-assembly and computation suggests that a shape can be considered the output of a self-assembly “program,” a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest self-assembly program that builds a shape and the shape's descriptional (Kolmogorov) complexity should be related. We show that when using a notion of a shape that is independent of scale, this is indeed so: in the tile assembly model, the minimal number of distinct tile types necessary to self-assemble a shape, at some scale, can be bounded both above and below in terms of the shape's Kolmogorov complexity. As part of the proof, we develop a universal constructor for this model of self-assembly that can execute an arbitrary Turing machine program specifying how to grow a shape. Our result implies, somewhat counterintuitively, that self-assembly of a scaled-up version of a shape often requires fewer tile types. Furthermore, the independence of scale in self-assembly theory appears to play the same crucial role as the independence of running time in the theory of computability. This leads to an elegant formulation of languages of shapes generated by self-assembly. Considering functions from bit strings to shapes, we show that the running-time complexity, with respect to Turing machines, is polynomially equivalent to the scale complexity of the same function implemented via self-assembly by a finite set of tile types. Our results also hold for shapes defined by Wang tiling—where there is no sense of a self-assembly process—except that here time complexity must be measured with respect to nondeterministic Turing machines.

Item Type:Article
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URLURL TypeDescription Paper
Soloveichik, David0000-0002-2585-4120
Winfree, Erik0000-0002-5899-7523
Additional Information:© 2007 Society for Industrial and Applied Mathematics. Received December 21, 2004; accepted June 30, 2006; published February 9, 2007. We thank Len Adleman, members of his group, Ashish Goel, and Paul Rothemund for fruitful discussions and suggestions. We also thank Rebecca Schulman and David Zhang for useful and entertaining conversations about descriptional complexity of tile systems, and an anonymous reviewer for a very careful reading of this paper and helpful comments. An extended abstract version of this work was previously published as Complexity of self-assembled shapes, in DNA Computing, Lecture Notes in Comput. Sci. 3384, Springer, Berlin, 2005, pp. 344–354. This work was supported by NSF CAREER grant 0093486.
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Subject Keywords:Kolmogorov complexity; scaled shapes; self-assembly; Wang tiles; universal constructor
Issue or Number:6
Record Number:CaltechAUTHORS:SOLsiamjc07
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8371
Deposited By: Archive Administrator
Deposited On:02 Aug 2007
Last Modified:08 Nov 2021 20:49

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